electroMechanicalLaw

A composite law that adds an active tension along a fibre direction to the stress of a passive law. The runtime type is:

electroMechanicalLaw

User Guide

What it computes

This is a finite-strain law: it implements the finite-strain update only, so both it and its passive law are used with a nonlinear geometry solid model.

The passive law, given in the passiveMechanicalLaw sub-dictionary, is evaluated first. The active stress is then added:

f0f0  = (f0/|f0|)*(f0/|f0|)
sigma = sigma_passive + symm(F & (Ta*f0f0) & F^T)/J

The active tension Ta is either a field or a ramped constant:

• with activeTensionFromField yes, Ta is the coupling input named by activeTensionFieldName (see Coupling inputs), and rampTime is not applied;
• otherwise Ta is activeTension, ramped linearly from zero:

Ta = (t/rampTime)*activeTension    for t < rampTime
Ta = activeTension                 otherwise, or when rampTime = 0

The density and the bulk modulus are those of the passive law. rho given on this law is passed down to the passive law when its own dictionary has none, so it can be given once, on the outer dictionary.

Model options

activeTensionFromField defaults to yes without an activeTension and to no with one. activeTension and rampTime are used only when activeTensionFromField is no. With the field on they are not used, and giving either prints a warning. Setting activeTensionFromField no without an activeTension is a fatal error, as is a negative rampTime.

The passiveMechanicalLaw sub-dictionary holds the passive law's type and all of its entries. Any finite-strain law can be used; the tutorials use GuccioneElastic.

Coupling inputs

With activeTensionFromField yes the law declares the active tension field (Ta by default) as a required scalar input. The manager gathers it on every evaluation, at cells, faces and boundary faces alike, and takes it from the first of:

1. a volScalarField of that name registered by another model, e.g. an electrophysiology model coupled to the solid;
2. the case directory named for it in inputCaseDirectories (or in the older TacaseDirectory spelling), read at the current time; see the framework README;
3. a file of that name in the current time directory, or failing that in 0.

If none exists the run stops with an error.

With activeTensionFromField no the field is not read, even if one is registered. In that case the manager prints a warning, once, if a volScalarField with the activeTensionFieldName name is registered, since the case is then running on the constant rather than on that field. An inputCaseDirectories entry for Ta while activeTensionFromField is no is a fatal error, because the law does not read that input; a TacaseDirectory entry is then ignored.

Fibre direction

f0 is a prescribed state variable. It is looked up as for GuccioneElastic: a volVectorField f0 in the current time directory, then in 0, then a registered field, and only if none exists the dictionary f0 (with uniformFibreField yes) or a zero vector. A zero-length fibre stops the run at the first evaluation. f0 is normalised before use.

This law and its passive law each hold their own f0. An f0 field on disk is read by both. uniformFibreField and f0 are not passed down, however: a passive GuccioneElastic with a uniform fibre needs them in its own passiveMechanicalLaw dictionary as well as here.

State variables

The passive law's state lives in a child state named passiveMechanicalLaw. Prescribed variables are neither written with the results nor restarted from; a passive law with persistent history has it written as described in that law's documentation, with restart files named <material>_<topology>_passiveMechanicalLaw_<variable>.

Tangents

• scalar and scalarDeviatoric: those of the passive law. The active tension is not included.
• fourthOrderFiniteDifference: recomputed by this law over the whole stress, active part included.
• fourthOrder (analytical): refused, since the passive law's tangent would miss the active term.

Mixed displacement-pressure formulation

The law provides the volumetric split exactly when its passive law does. The volumetric response is the passive law's dU/dJ; the active stress, whose mean is not a function of J, stays in the isochoric part that the solved pressure does not replace. So with a passive GuccioneElastic the law can be used with a mixed formulation, as in LandEtAl2015/problem3 run with ./Allrun pressure.

Differences from the legacy law

• A registered Ta field is no longer picked up automatically. Set activeTensionFromField yes to use one; without it the constant is used and a warning is printed if a Ta field is registered. The field name can be changed with activeTensionFieldName.
• Ta can now also be read from a file in the time directory, or from another case directory with inputCaseDirectories.
• f0 may be given in this dictionary with uniformFibreField, is normalised, and no longer needs a face field f0f.
• The legacy face-centred path multiplied the active term by J rather than dividing by it; every evaluation path now uses the relation above.
• The fourth-order finite-difference tangent includes the active stress; the legacy law returned the passive law's tangent.
• regionName is no longer read. solvePressureEqn is read by the solid model wherever it appears in the material's dictionary, including inside passiveMechanicalLaw, and applies to the whole mesh; see the material models page.

Example

From the LandEtAl2015/problem3 tutorial (comments and unused entries removed), where f0 is written into 0 by setFibreField:

planeStress     no;

mechanical
(
    heartTissue
    {
        type            electroMechanicalLaw;
        rho             rho [1 -3 0 0 0 0 0] 3000;
        activeTension   activeTension [1 -1 -2 0 0 0 0] 60.0e3;
        rampTime        1.0;

        passiveMechanicalLaw
        {
            type            GuccioneElastic;
            k               k [1 -1 -2 0 0 0 0] 2e3;
            cf              8.0;
            ct              2.0;
            cfs             4.0;
            bulkModulus     bulkModulus [1 -1 -2 0 0 0 0] 16e6;
        }
    }
);

To drive the tension from a field supplied by another model, or from files Ta in the time directories, add:

        activeTensionFromField  yes;
        // activeTensionFieldName  Ta;

Tutorials

• solids/hyperelasticity/LandEtAl2015/problem3